Clifford algebras and Spin groups

As a provisional definition, clifford algebraover a field can be defined as

where,

Easy to see that has dimension . can also be thought of as

where as vector space, is the tensor algebra, is same as above, except that ‘s are standard basis for . This observation leads to a more general definition of clifford algebra, where is a vector space equipped with a symmetric bilinear form

Definition 1 Let be a vector space with symmetric biliear form and quadratic form . Then the clifford algebraover can be defined as

Remark 1These are some of the properties that enjoys

There is a natural inclusion of .

If then the exterior algebra

Let denote the clifford multiplication( induced by the tensor product of ), then

Universal Property : Let , where is a -algebra, such that , then there exists an unique map such that , that is the following diagram commutes

A map , such that , extends to a -algebra homomorphismThus the orthogonal group has an action on

Proposition 2 is a filtered algebra whose associate graded is

Before proving the theorem, recall the following definition

Definition 3If is a -algebra then a filtration of is sequence of subspaces

The associate gradedof is defined as

Proof: Let be the quotient map

Define, ( fold tensor product). Define filtration on by setting

Define filtration be the filtration on the clifford algebra. Note . Hence, in the associated graded . On the other hand the relation prevails in the associated graded. Hence the associate graded is isomorphic to .

Remark 2 is a -graded algebra.

Definition 4Recall,. Define,

and

On we have an involution map, which is induced by the involution on given by,

If then

Let and , then observe

Lemma 5There exist short exact sequences

and

where is the map which sends

Let be a field. Recall, tensor product of -algebras and is a -algebra, denoted by and multiplication is given by

moreover if denotes the set of all matrices. Then we have the following isomorphism

Define

Remark 3If , then

This follows from the fact that the quadratic forms and induces isomorphic innerproduct structure on where the isomorphism sends

Theorem 6If , then we have the following isomorphisms

Corollary 7(Bott Periodicity) As a consequence of (iv) we have

if even, and

if is odd.

To work out the case when the underlying field is . For any field we have the following isomorphisms.

Lemma 8For any field

Proof: Let denote the standard basis of and cannonical generatoring set of the Clifford algebra.

To get the first isomorphism we simply produce a map given by sendingandIt is easy to check that the above map is an isomorphism.

is similar to .

One can explicitly check some of the lower dimension cases( ). Then one can repeatedly use the isomorphisms in previous lemma. One has to work upto dimension when , before one sees the patern, which is called the Bott periodicity. TSome of the calculations are as follows calculations are as follows

In general one gets,

Putting all these observations together we get

Theorem 9 The Bott periodicity in case of real number looks like

Lemma 10 As -algebras Proof:The isomorphism is given explicitly by the map induced by sending